Naber G.L. The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity

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224 4 Prologue and Epilogue: The de Sitter Universe

=sinh



sin

cos

+sin

sin

cos

+sin

sin

θ +cos



− cosh

=sinh

− cosh

= −1

=(χ

,χ

)=0−cosh

sin

sin φ

cos φ

sin θ cos θ

+cosh

sin

sin φ

cos φ

sin θ cos θ

Exercise 4.3.17 Compute the rest and show that the only nonzero

,i,j=1, 2, 3, 4, are

=cosh

sin

=cosh

sin

= −1

Notice that the restriction of the M

inner product does, indeed, deﬁne a

Lorentz metric on D since, at each point p ∈D, there is a basis e

for the tangent space T

(D) satisfying g(e

)=η

,i,j=1, 2, 3, 4. Indeed,

one can take e

= χ

(p)andlete

be the normalized versions of

,χ

, i.e.,

cosh t

(p)

cosh t

sin φ

(p)

cosh t

sin φ

(p).

With this Lorentz metric, D is called the de Sitter spacetime and will be

denoted dS.

Before recording more examples we will introduce a more traditional and

generally more convenient means of displaying the metric components. We

will illustrate the idea ﬁrst for the 2-sphere S

. To facilitate the notation we

will (temporarily) denote the standard spherical coordinates φ and θ on S

by x

and x

, respectively.

χ :(0,π) × (0, 2π) −→ S

χ(x

)=(sin(x

)cos(x

), sin(x

)sin(x

), cos(x

))

4.3 Mathematical Machinery 225

Let α :[a, b] → S

be a smooth curve in S

given by

α(t)=χ(x

(t),x

(t))

=(sin(x

(t)) cos(x

(t)), sin(x

(t)) sin(x

(t)), cos(x

(t))).

For each t in [a, b], the Chain Rule gives



(t)=

(t),x

(t)) +

(t),x

(t))

(t),x

(t)).

The Riemannian metric g we have deﬁned on S

allows us to compute the

squared magnitude of α



(t) as follows.

g(α



(t),α



(t)) = g





g(χ

,χ

)

= g

The square root of g(α



(t),α



(t)) is the curve’s “speed” which, when inte-

grated from a to t gives the arc length s = s(t). Consequently,





= g

which it is customary to write more succinctly in “diﬀerential form” as

= g

Reverting to φ and θ and substituting the values of g

that we have computed

((4.3.17)) gives

= dφ

+sin

φdθ

. (4.3.18)

This is generally called the line element of S

. We regard it as a horizontal

display of the metric components g

=1,g

=sin

φ, g

= g

=0anda

convenient way to remember how to compute arc lengths.

Remark: The symbols in (4.3.18) can all be given precise meanings in the

lang

uag

e of diﬀerential forms, but we will have no need to do so.

The same notational device is employed for any Riemannian or Lorentz

metric. For example, writing x

= x, x

= y, x

= z for the standard

coordinates on R

one has

= dx

+ dy

+ dz

226 4 Prologue and Epilogue: The de Sitter Universe

= g

=1andg

=0fori, j =1, 2, 3,i= j), whereas, in spherical

coordinates on R

= dρ

+ ρ

(dφ

+sin

φdθ

). (4.3.19)

For R

3,1

it would be

=(dx

)

+(dx

)

+(dx

)

− (dx

)

, (4.3.20)

while for S

with spherical coordinates φ

,φ

,θ,

= dφ

+sin



dφ

+sin

dθ



. (4.3.21)

Finally, for de Sitter spacetime in global coordinates, Exercise 4.3.17 gives

=cosh



dφ

+sin



dφ

+sin

dθ



− dt

. (4.3.22)

Exercise 4.3.18 Show that the line element for de Sitter spacetime, written

in conformal coordinates (φ

,φ

,θ,t

)is

cos



dφ

+sin



dφ

+sin

dθ



− dt



Exercise 4.3.19 Show that the line element for de Sitter spacetime, written

in planar coordinates (x

)isds

= e

((dx

)

+(dx

)

(dx

)

) − dt

, or, using spherical coordinates for x

= e

(dρ

+ ρ

(dφ

+sin

φdθ

)) − dt

Exercise 4.3.20 Show that the line element for de Sitter spacetime, written

in hyperbolic coordinates (φ, θ, ψ, t

)is

=sinh

(dψ

+sinh

ψ(dφ

+sin

φdθ

)) − dt

One should notice, however, that in the case of dS (or any other Lorentz

manifold) the interpretation of the line element requires some care. Just as

in Minkowski spacetime, a curve in dS might well have a velocity vector that

is null at each point so that g

is zero everywhere and its “arc length”

is zero. If the velocity vector is timelike everywhere, then





= g

would make

pure imaginary. To exercise the proper care we mimic our

deﬁnitions for M.

If M is a spacetime, then a smooth curve α : I → M is said to be spacelike,

timelike,ornull if its tangent vector α



(t)satisﬁesg

α(t)

(α



(t),α



(t)) > 0,

α(t)

(α



(t),α



(t)) < 0, or g

α(t)

(α



(t),α



(t)) = 0 for each t ∈ I.IfI has

an endpoint t

we require that these conditions be satisﬁed there as well in

the sense that any smooth extension of α to an open interval about t

has a

tangent vector at t

satisfying the required condition. Notice that in dS this

4.3 Mathematical Machinery 227

simply amounts to the requirement that each α



(t), regarded as a vector in

is spacelike, timelike or null in M

. We will say that a timelike or null

curve in dS is future-directed (resp., past-directed )ifeachα



(t), regarded as

a vector in M

, is future-directed (resp., past-directed).

Remark: The notion of a spacetime, as we have deﬁned it, is very general

and there are examples in which it is not possible to deﬁne unambiguous

notions of future-directed and past-directed (see [HE], page 130). This is

essentially a sort of “orientability” issue analogous to the fact that, for some

surfaces in R

such as the M¨obius strip, it is impossible to deﬁne a continuous

nonzero normal vector ﬁeld over the entire surface. We care only about M

and dS and so the issue will not arise for us.

A future-directed timelike curve in dS is called a timelike worldline and we

ascribe to such a curve the same physical interpretation we did in M (the

worldline of some material particle). Just as in M one can deﬁne the proper

time length L(α)ofsuchacurveα :[a, b] → dS by

L(α)=





−(α



(t),α



(t)) dt

and a proper time parameter τ = τ(t)alongα by

τ = τ(t)=





−(α



(u),α



(u)) du.

We will go even further and induce causality relations on dS from those on

. Speciﬁcally, for distinct points x and y in dS we will deﬁne x  y if and

only if y − x is timelike and future-directed in M

and x<yif and only if

y − x is null and future-directed in M

Remark: Although we will have no need of the result, we point out that,

with these deﬁnitions, Zeeman’s Theorem 1.6.2 remains true in dS in the

following sense. A bijection F : dS → dS that preserves < in both directions

also preserves  in both directions and is, in fact, the restriction to dS of

some orthochronous orthogonal transformation of M

(see [Lest]).

Let us think for a moment about the sort of timelike curve in dS that

should model the worldline of a material particle that is “free”, i.e., in free

fall. As we saw in Section 4.2, at each point on such a worldline there is a

local inertial frame (freely falling elevator) from which it can be observed

and that, relative to such a frame, the worldline will appear (approximately)

“straight.” Curves in a manifold with (Riemannian or Lorentz) metric that

are “locally straight” are called geodesics. There are various approaches to the

formulation of a precise deﬁnition, but we will follow a path that is adapted

to the simplicity of the examples we wish to consider. The motivation for our

approach is most easily understood in the context of smooth surfaces in R

so we will ﬁrst let the reader work through some of this.

228 4 Prologue and Epilogue: The de Sitter Universe

Exercise 4.3.21 Let M be a smooth surface (i.e., 2-manifold) in R

with

Riemannian metric g obtained by restricting the R

-inner product  ,  to each

tangent space. Let χ : U → M ⊆ R

be a coordinate patch for M and let

α = α(t)beasmoothcurveinM whose image is contained in χ(U ). At each

t the tangent vector α



(t) is, by deﬁnition, in T

α(t)

(M), but the acceleration



(t) in general will not lie in T

α(t)

(M) since it will have both tangential and

normal components, i.e.,



(t)=α



tan

(t)+α



nor

(t),

where α



tan

(t) ∈ T

α(t)

(M)andα



nor

(t),v =0foreveryv ∈ T

α(t)

(M).

(a) Write α(t)=χ(x

(t),x

(t)) and show that



(t)=

(t),x

(t)) +

(t),x

(t)),

where

∂

∂x



∂

∂x

∂

∂x

∂

∂x



(b) Show that the cross product χ

× χ

is nonzero at each point of χ(U)

and so N = χ

× χ

/χ

× χ

 is a unit normal vector to each point

of χ(U). Note: This normal vector ﬁeld generally exists only locally on

χ(U). There are surfaces (such as the M¨obius strip) on which it is not

possible to deﬁne a continuous, nonvanishing ﬁeld of normal vectors.

into tangential and normal components to obtain

=Γ

+ L

where χ

,χ

 =Γ

and L

= χ

,N.

(d) Deﬁne Γ

r,ij

= g

and show that

∂g

∂x

=Γ

i,jk

+Γ

j,ik

. (4.3.23)

(e) Denote by (g

) the inverse of the matrix (g

)andshowthat



∂g

∂x

∂g

∂x

−

∂g

∂x



. (4.3.24)

Hint: Permute the indices ijkin (4.3.23) to obtain expressions for each

the d

erivatives in (4.3.24) and combine them using the symmetries

i,jk

=Γ

i,kj

, i,j,k =1, 2.

4.3 Mathematical Machinery 229

(f) Conclude that



tan

(t)=



+Γ

(t),x

(t))



(t),x

(t))

and



nor

(t)=L

(t),x

(t)) N(x

(t),x

(t)).

If, in Exercise 4.3.21, t = s is the arc length parameter for α,t

nα



(s)is

the curvature vector of α. One regards α



nor

as that part of α’s curvature that

it must possess simply by virtue of the fact that it is constrained to remain

in M,whereasα



tan

is the part α contributes on its own by “curving in M .”

Curves with α



tan

= 0 are thought to “curve only as much as they must to

remain in M ” and so are the closest thing in M to a straight line (we will

see shortly that in S

these are just constant speed parametrizations of great

circles, i.e., intersections of S

with planes through the origin in R

There is only one obstacle to carrying out the entire calculation in

Exercise 4.3.21 for any χ(U)ona

nn-man

ifold M in R

and that is ex-

istence of the unit normal ﬁeld N.InR

,m>3, there is no natural

concept of a cross product and, in any case, the tangent space T

(M)is

not spanned by just two vectors. For the simple examples of interest to us,

however, the obstacle is easily overcome. At each point p on the n-sphere

⊆ R

n+1

, for instance, the vector p in R

n+1

is itself a unit normal vector

to T

). Indeed, if α is any smooth curve in S

through p at t = t

,then

α(t)=(u

(t),...,u

n+1

(t)) implies

(t))

+ ···+(u

n+1

(t))

(t)

+ ···+2u

n+1

(t)

n+1

which, at t = t

,gives

p, α



) =0.

Similarly, if p ∈ dS ⊆M

and α(t)=(u

(t), ··· ,u

(t)) is a smooth curve in

dS with α(t

)=p,then

(t))

+ ···+(u

(t))

− (u

(t))

gives

(p, α



)) = 0

so p is a unit normal vector to T

(dS ) (“unit” and “normal” now refer to the

Lorentz metric of dS,ofcourse).

230 4 Prologue and Epilogue: The de Sitter Universe

In either of these cases the calculations in Exercise 4.3.21 (with N replaced

by N(p)=p) can be repeated verbatim to resolve the acceleration α



(t)of

a smooth curve into tangential and normal components with



tan

(t)=



+Γ



(4.3.25)

in any coordinate patch, where



∂g

∂x

∂g

∂x

−

∂g

∂x



(4.3.26)

(these are called the Christoﬀel symbols of the metric in the given coordinate

system).

Those curves for which α



tan

(t)=0foreacht are called geodesics and they

satisfy

+Γ

= 0 (4.3.27)

in any coordinate patch.

Remark: Geodesics can be introduced in many ways and in much more

general contexts, but the end result is always the system (4.3.27)ofordinary

diﬀere

ial equations. Notice that equations (4.3.27) are trivially satisﬁed by

con

stant curve α(t)=p ∈ M. Even though such constant curves are not

smooth in our sense we would like them to “count” and so we will refer to

them as degenerate geodesics.

To get some sense of the complexity of the equations (4.3.27) we should

write

them

out in the case of most interest to us. Thus, we consider de Sitter

spacetime dS in global coordinates x

= φ

= θ, x

= t

.From

Exercise 4.3.17,

) = diag(cosh

, cosh

sin

, cosh

sin

, −1)

so (g

) is the diagonal matrix whose entries are the reciprocals of these.

Because both of these are diagonal and independent of x

= θ,theChristoﬀel

symbols (4.3.26) simplify a bit to



∂g

∂x

∂g

∂x

−

∂g

∂x



where there is no sum over r,allx

-derivatives vanish and all g

with k = l

are zero. We compute a few of these.



0+0−

∂g

∂x



(−1)(−2cosht

sinh t

)

=cosht

sinh t

4.3 Mathematical Machinery 231



0+0−

∂g

∂x





cosh



(cosh

(−2sinφ

cos φ

))

= −sin φ

cos φ



∂g

∂x

− 0





cosh

sin





2cosht

sinh t

sin



sinh t

cosh t



∂g

∂x

+0− 0





cosh

sin



cosh

(2 sin φ

cos φ

)

cos φ

sin φ

Exercise 4.3.22 Show that the only nonvanishing Christoﬀel symbols for

dS in global coordinates are as follows.

=cosht

sinh t

=cosht

sinh t

sin

=cosht

sinh t

sin

=Γ

sinh t

cosh t

,i=1, 2, 3

= −sin φ

cos φ

= −sin φ

cos φ

sin

= −sin φ

cos φ

=Γ

cos φ

sin φ

=Γ

cos φ

sin φ

=Γ

With these one can write out the geodesic equations (4.3.27)forr =1, 2, 3, 4.

For

ample, if r =1,

+Γ

becomes

+Γ





+Γ





+2Γ

=0,

232 4 Prologue and Epilogue: The de Sitter Universe

− sin φ

cos φ



dφ



− sin φ

cos φ

sin



dθ



sinh t

cosh t

dφ

=0.

Exercise 4.3.23 Write out the r =2, 3, and 4 equations to obtain a system

of four coupled second order ordinary diﬀerential equations for the geodesics

of dS.

The problem of explicitly solving the geodesic equations can be formidable.

However, a few basic facts about systems of ordinary diﬀerential equations

(and some inspired guesswork) will relieve us of this burden. For instance,

(4.3.27) is nothing more that a system of second order ordinary diﬀerential

equa

tio

ns f

or the functions x

(t) so that standard existence and uniqueness

theorems for such systems imply that, for any given initial position α(t

)and

initial velocity α



) there is a unique solution α(t)deﬁnedonsomeinterval

about t

satisfying these initial conditions. More precisely, one obtains the

Existence and Uni queness Theorem: Let M be a manifold with

(Riemannian or Lorentzian) metric g and ﬁx some t

∈ R. Then for any

p ∈ M and any v ∈ T

(M) there exists a unique geodesic α

: I

→ M

such that

1. α

)=p, α



)=v, and

2. the interval I

is maximal in the sense that if α : I → M is any geodesic

satisfying α(t

)=p and α



)=v, then I ⊆ I

and α = α

| I.

We will ﬁnd the uniqueness asserted in this result particularly useful since

it assures us that if we have somehow managed to conjure up geodesics in

every direction v at p, then we will, in fact, have all the geodesics through p.

We will apply this procedure to a few examples shortly (e.g., by “guessing”

that the geodesics of S

should be great circles). First, however, we must

come to understand that a geodesic is more than its image in M,whichmust

be parametrized in a very particular way if it is to satisfy (4.3.27). First we

show

at a given geodesic can be reparametrized in only a rather trivial way

if it is to remain a geodesic.

Lemma 4.3.1 Let M be a manifold with (Riemannian or Lorentzian) metric

gandletα : I → M be a nondegenerate geodesic. Suppose J ⊆ R is an

interval and h : J → I, t = h(s), is a smooth function with h



(s) > 0 for

each s ∈ J. Then the reparametrization

β = α ◦ h : J −→ M

of α is a geodesic if and only if h(s)=as + b for some constants a and b.

4.3 Mathematical Machinery 233

Proof: The Chain Rule gives



(s)=α



(h(s)) h



(s)

and



(s)=α



(h(s))(h



(s))

+ α



(h(s)) h



(s)



tan

(s)=α



(h(s)) h



(s)

(α is a geodesic). Thus, β is a geodesic if and only if α



(h(s))h



(s) = 0. Since

α is a nondegenerate geodesic, α



is never zero (otherwise uniqueness would

imply that α is the constant curve). Thus, h



(s) = 0 for every s in J so

h(s)=as + b for some constants a and b. 

We can say much more about the parametrizations of a geodesic, however.

We will now prove that geodesics are always constant speed curves.

Theorem 4.3.2 Let M be a manifold with (Riemannian or Lorentzian) met-

ric g. Suppose α : I → M is a geodesic. Then g(α



(t),α



(t)) is constant on I.

Proof: We will show that

g(α



(t),α



(t)) = 0 at each t in I. It will clearly

suﬃce to focus our attention on some subinterval of I that maps into χ(U)for

some coordinate patch χ : U → M ⊆ R

.Writingα(t)=χ(x

(t),...,x

(t))

we have

(g(α



(t),α



(t))) =



(t),...,x

(t))



To simplify the notation we will drop the arguments x

(t),...,x

(t)and

compute





= g

+ g





=2g

∂g

∂x

(4.3.28)

Now,

= −Γ

since α is a geodesic so

2 g

= −2 g

Moreover,



∂g

∂x

∂g

∂x

−

∂g

∂x





∂g

∂x

∂g

∂x

−

∂g

∂x





∂g

∂x

∂g

∂x

−

∂g

∂x

